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Subspace Algorithm

Subspace Algorithm

Subspace Algorithm

(OP)
I'm trying to understand subspace algorithms. I have read a number of references, but still have a question.

For a free vibration problem, with no noise,

x(k+1)=Ax(k)
y(k) = Cx(k)

k = 1 to S

for
yhat(k,N) = [y(k), y(k+1), y(k+2), ... y(k+N-1)]
Y = [yhat(1); yhat(2); yhat(3); ...;yhat(M)]

S = M + N - 1


Its easy to show that

Y = Gamma xhat(1,N)

Gamma = [C; CA; CA^2; ...; CA^(M-1)]

and Gamma can be calculated from singular values of Y

Y = P Sigma transpose(V)

then

Gamma = P
where the dimension of P can be reduced based on a plot of the singular values.

I understand the math up to this point

but why does

x(1) = sum ( Sigma(i,i) * V(:,i)) ?

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